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Power series for the number e

The number e = 2.7182…can be expressed as the power series

 


Explanation

We start with the function

Substitution in the definition of the derivative gives

Evaluation shows

For x = 0 you get

This is a constant, as it does't contain any x variable. The value only depends on a, the base. So you can write

Now we want to know for what value of a the function f ′(0) = 1, then in this case f ′(x) = ax, and the function is its own derivative. So we calculate

Multiplication by h

Reshuffle

Exponentiation

Simplification

Set , and because → 0 this leads to → ∞ , and so

We use the binomium development, and obtain

This was a limit, so we write

If we apply L'Hôpitals rule several times on the different terms, we get

The subsequent terms yield

1     = 1.0000
1/1! = 1.0000
1/2! = 0.5000
1/3! = 0.1667
1/4! = 0.0417
1/5! = 0.0083
1/6! = 0.0014
           -------
  e   = 2.718…

The result is correct for 3 digits.

 


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